Theorem xpdom2g 6502

Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
xpdom2g	|- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )

Proof of Theorem xpdom2g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 4427	. . . . 5 |- ( x = C -> ( x X. A ) = ( C X. A ) )
2		xpeq1 4427	. . . . 5 |- ( x = C -> ( x X. B ) = ( C X. B ) )
3	1, 2	breq12d 3835	. . . 4 |- ( x = C -> ( ( x X. A ) ~<_ ( x X. B ) <-> ( C X. A ) ~<_ ( C X. B ) ) )
4	3	imbi2d 228	. . 3 |- ( x = C -> ( ( A ~<_ B -> ( x X. A ) ~<_ ( x X. B ) ) <-> ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) ) )
5		vex 2618	. . . 4 |- x e. _V
6	5	xpdom2 6501	. . 3 |- ( A ~<_ B -> ( x X. A ) ~<_ ( x X. B ) )
7	4, 6	vtoclg 2672	. 2 |- ( C e. V -> ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) )
8	7	imp 122	1 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1287   e. wcel 1436   class class class wbr 3822   X. cxp 4411   ~<_ cdom 6410

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fv 4991  df-dom 6413

This theorem is referenced by:  xpdom1g  6503